Adiabatic and post-adiabatic approaches to extreme mass ratio inspiral

0.62 MB·

by  Scott A. Hughes

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Adiabatic and post-adiabatic approaches to extreme mass ratio inspiral
Scott A.Hughes∗
Department of Physics and MIT Kavli Institute, MIT, Cambridge, MA 02139, USA
∗E-mail: [email protected]
Extrememass ratioinspirals(EMRIs)show astrong separationof timescales, withthe
time characterizing inspiral, Ti, much longer than any time To characterizing orbital
6 motions. The ratioof these timescales (which isessentially an EMRI’smass ratio) can
1 be regarded as a parameter that controls a perturbative expansion. Here we describe
0 thevalueandlimitationsofan“adiabatic”descriptionofthesebinaries,whichusesonly
2 the leading terms arisingfrom such a two-timescale expansion. An adiabatic approach
breaks down when orbits evolve through resonances, with important dynamical and
n
observationalconsequences. Wedescribetheshortfallsofanapproachthatonlyincludes
a
theadiabaticcontributionstoEMRIevolution,andoutlinewhatmustbedonetoevolve
J
these systems through resonance and to improve our ability to model EMRI systems

8 moregenerally.

] Keywords:Blackholes;blackholeperturbationtheory;gravitationalwaves
c
q
-
r 1. Motivation: The large-mass ratio limit of the two-body problem
g
and extreme mass ratio inspirals
[

1 Binary systems in which one body is much more massive than the other can be
v analyzed perturbatively. We can describe such a binary as an exact black hole
2
solution of general relativity (corresponding to the larger member of the binary)
4
0 plus a correction due to the smaller body. Because the perturbation equations are
2 much simpler to solve than the complete equations of general relativity, this turns
0
out to be a limit that can be modeled very accurately and precisely.
.
1 Atleasttwomajorsciencegoalsdrivestudiesoflargemassratiosystems. First,
0
thesebinariesrepresentalimitofthetwo-bodyproblemthatcanbesolvedwithhigh
6
1 precision. Assuch,thestudyofthesebinariesprovidesimportantinputtoprograms
: tosolvethetwo-bodyproblemofgeneralrelativitymoregenerally,suchasnumerical
v
i relativity and the effective one-body approach1–3. Second, astrophysical extreme
X
mass ratio inspirals (EMRIs) are expected to be important sources for space-based
r GW detectors such as eLISA4 and DECIGO5.
a
In this article, we will focus on the role of EMRIs as sources of gravitational
waves(GWs). Suchbinariesarecreatedwhenmultibodyinteractionsscatterstellar
mass compact objects onto a strong-field, relativistic orbit of the black hole in a
galaxy’scenter. Further evolutionisthendrivenbyGW emission. Ifthe blackhole
hasamassofaround105−107M⊙,thenthesearetargetsforadetectorlikeeLISA.
The GWs that they generate can be heard out to z ∼ 0.5−1; we expect dozens
to hundreds of events over a space-based detector’s mission lifetime6. Measuring
the GWs from these events will provide precisiondata on the characteristics of the
large black hole, on the small body’s orbit, and on the mass of the small body —
in short, a precision probe of the astrophysical population of galactic center black
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holes,andinformationaboutthe populationofstarsinthe centersofgalaxies. It is
expected thatEMRI waveswillevenbe preciseenoughto testthe Kerrsolutionby
mapping the multipolar structure of the dense, dark objects in galactic cores that
we presume to be general relativity’s black holes.
2. Modeling EMRIs
To achieve these science goals,we must accurately model EMRI waves. How accu-
rate must ourmodels be? The answerdepends onthe purpose to whichwe put the
model7. The instantaneous EMRI wave amplitude will typically be about a factor
of10smallerthandetectornoise. Byfittingtoamodeltemplatethatiscoherentin
phase with the data for N cycles, we (roughly speaking) boost the signal-to-noise
ratio (SNR) by N1/2. For detection purposes (determining that a signal is in your
data) your model should hold phase with the signal to within ∆φ . 1 radian over
thesignal’sdurationa. Thebestfitislikelytohavelargesystematicerrors,butthat
is acceptable if our goal is just to establish that a signal is present. For measure-
ment purposes (e.g., using the detected wave to determine source parameters), our
model must be accurate enough that systematic errors (due to inadequate model-
ing) are smaller than statistic errors (due to noise). A crude rule of thumb is that
the template’s phase must match the signal to within ∆φ.1/SNR.
Turn now to an overview of how one makes an EMRI model. We will use the
action-angleapproachdescribedbyFlanaganandHinderer8 todescribethemotion
of the small body m in the spacetime of a larger black hole of mass M:
dq
α =ω (J)+εg (q ,q ,J)+O(ε2), (1)
α α θ r
dλ
dJ
i =εG (q ,q ,J)+O(ε2). (2)
i θ r
dλ
In these equations, λ is a time variable that is well adapted to strong-field Kerr
orbits, and ε=m/M. The angle variables
.
q =(q ,q ,q ,q ) (3)
α t r θ φ
eachdescribe the motion ofthe small body aboutthe black hole in suitable coordi-
nates; the action variables
.
J =(E/m,L /m,Q/m2) (4)
i z
correspondtointegralsofthemotionthatareconservedalonga“background”orbit
(i.e., in the limit of purely geodesic motion). We will examine the forcing terms g
α
and G in more detail momentarily.
i
aNote that “duration” does not necessary mean the complete span of the signal in your data.
Onecanbreakthedataintosegments,coherentlyintegrateeachsegmentagainstatemplate,and
combineeachprocessedsegmentintoasinglestatistic.
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To understand the small body’s motion in this framework, let us examine Eqs.
(1) and (2) more carefully. At zeroth order in ε, these equations become
dq dJ
α =ω (J), i =0. (5)
α
dλ dλ
In other words, when the O(ε) corrections to the equations of motion are not in-
cluded, the angle variables accumulate at a rate set by their associated frequency,
and the integrals of the motion are constant. Equation (5) expresses the fact that
the motion at zeroth order in the small body’s mass is a Kerr geodesic.
Whenwegotothenextorderinε,theforcingtermsg andG mustbeincluded:
α i
dq dJ
α =ω (J)+εg (q ,q ,J), i =εG (q ,q ,J). (6)
α α θ r i θ r
dλ dλ
Thesetermspushthesmallbodyawayfromthegeodesic,andconstitutetheleading
self force correction to the small body’s motion.
3. The two-timescale expansion
Further insight into EMRI evolution can be found by separating the forcing terms
into their averages and oscillations about the average:
G (q ,q ,J)=hG (J)i+δG (q ,q ,J), (7)
i θ r i i θ r
where
1 2π 2π
hG (J)i = dq dq G (q ,q ,J), (8)
i (2π)2 Z θZ θ i θ r
0 0
δG (q ,q ,J) =G (q ,q ,J)−hG (J)i. (9)
i θ r i θ r i
Weapplyasimilarsplittog (q ,q ,J). Rewritetheequationsofmotiononcemore:
α θ r
dq
α =ω (J)+εhg (J)i+εδg (q ,q ,J), (10)
α α α θ r
dλ
dJ
i =εhG (J)i+εδG (q ,q ,J). (11)
i i θ r
dλ
The averaged forcing term hG (J)i describes the leading evolution of the small
i
body’s integrals of motion; its components describe the dissipative evolution of E,
L ,andQ. Thistermdrivesthesecularevolutionofthesystem’sorbitalparameters
z
on an inspiral time scale T ∼M/ε=M2/m. The averagedforcing term hg (J)i is
i α
equivalent to a shift of the frequencies:
ω (J)−→ω (J)+εhg (J)i. (12)
α α α
This shift is the leading conservative contribution of the small body’s self force.
These forcing terms are nearly constant, varying on the long timescale T ∼M2/m
i
that characterizes the rate of change of the integrals of motion J. By contrast, the
forcing terms δG (q ,q ,J) and δg (q ,q ,J) vary rapidly on a timescale T ∼ M
i θ r α θ r o
that characterizes the small body’s orbital motion. Their impact is (usually) much
less important than the impact of the averagedterms hG (J) and hg (J)i.
i α
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Asdiscussedabove,themostimportantdetailweneedtounderstandtocharac-
terize models for GW measurements is the phaseb accumulated over some interval:
t2
Φ(t1,t2)=Z ω(t)dt=Φdiss−1+Φcons−1+Φdiss−2+Φcons−2+... (13)
t1
The contributions to this phase have the following scalings with masses, and arise
from the following pieces of source physics:
• Φdiss−1 = O(Ti/To) = O(M/m): The slowing evolving geodesic frequency
ω =1/T ∼1/M integrated over the inspiral time T ∼M2/m.
o i
• Φcons−1 =O(ǫTi/To)=O(1): The conservativecorrectionto the frequency
δω ∼εω integrated over the inspiral time T.
i
• Φdiss−2 = O(ǫTi/To) = O(1): The slowly evolving geodesic frequency ω
integrated against the oscillatory correction to the inspiral time δT ∼εT.
i i
• Φcons−2 = O(ǫ2Ti/To) = O(m/M): The conservative correction to the fre-
quencyδω ∼εω integratedagainstthe oscillatorycorrectiontotheinspiral
time δT ∼εT.
i i
These schematic countings suggestthat we must include the leading adiabatic con-
servative piece and perhaps the first oscillatory dissipative piece in order to have
effective detection templates. We certainly will need to go farther for measurement
purposes, or else just accept that a certain level of systematic error may be very
difficult to remove from EMRI waveformmodels.
4. Adiabaticity and its limitations: Resonant orbits
The fact that the oscillatory contributions to the EMRI model are subleading sug-
gests that a useful approximation may be to ignore them at first pass. Doing so
gives us the adiabatic approximation to EMRI evolution:
dq dJ
α =ω (J)+εhg (J)i, i =εhG (J)i. (14)
α α i
dλ dλ
Formostblackholeorbits,thisapproximationworkswell. Thisisbecausethesmall
body’smotiontypicallyisergodic: afterasmallnumberoforbits(requiringfarless
than the inspiral time), the small body has come close to every point it is allowed
to pass through. Its motion thus averages the forcing terms in the equations of
motion more or less automatically. This is illustrated in the left-hand panel of Fig.
1 (adapted from Fig. 1 of Ref. 9), which shows the path in (r,θ) traced out by an
orbit. Roughly nine radial orbits are shown here. Given enough time, this trace
would fill the entire (r,θ) plane over the domain 2M .r .12M, 70◦ ≤θ ≤110◦.
However,there existorbitsfor whichthis averagingdoes notoccur. Ifthe small
body’s θ and r frequencies are in a low-order resonance, then the motion is not
bBear in mind that the following equation is meant to be schematic. The frequency ω(t) which
appearsunderthisintegralisinfactaharmonicofthevariousfrequencieswhichcharacterizeKerr
blackholeorbits,andassuchhasmorecontributions thanareindicatedinthissketch.
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ergodic, but instead traces out a Lissajous figure in the (r,θ) plane. An example is
shown in the right-hand panel of Fig. 1. Because these orbits do not come close to
allallowedpointsintheaccessiblephysicalspace,theydonoteffectivelyaveragethe
forcing functions g (q ,q ,J) and G (q ,q ,J). Indeed, different initial conditions
α θ r i θ r
trace outdifferent Lissajousfigures. This meansthat the detailedmanner inwhich
averaging fails depends on an orbit’s phase as it enters resonance.
Fig. 1. Three example Kerr black hole orbits. All oscillate between 2M .r .12M and 70◦ ≤
θ≤110◦;roughly9radialcyclesareshownforeachcase. Theleft-handpanelshowstheseorbits
forablackholewithspinparametera=0.95M. Thismotionisergodic—givenenoughtime,the
orbit would come arbirarilyclose to every accessible point in (r,θ). The right-hand panel shows
twoorbitsforspina=0.9M. Theθandrmotionsoftheseorbitsareina3:1resonance: theorbit
oscillates three times in θ over each radial cycle. No matter how long we wait, the motion will
be confined to the Lissajous figures shown here. The particular points the orbit passes through
dependonthesystem’sphaseasitentersresonance. Twoexamples areshownhere.
To make this more quantitative, examine the Fourier expansion of G :
i
G (q ,q ,J)= G (J)ei(kqθ+nqr) . (15)
i θ r i;kn
X
kn
Foranon-resonantorbit,onlyonetermontheright-handsurvivesaveraging: using
Eq. (8), we have
hGi(J)inon−res =Gi;00(J). (16)
since hei(kqθ+nqr)i=0 except for k =n=0.
This is not the case for resonant orbits. When Ω /Ω are in a ratio of small
θ r
integers, then q /q are also in a ratio of small integers. We now find that many
θ r
terms on the right-hand side of Eq. (15) survive the average:
hG (J)i =G (J)+ G . (17)
i res i;00 i;kn
X
(k,n)
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The final sum is over all pairs (k,n) that satisfy kq +nq =0.
θ r
Averaging,and thus the adiabatic approximation,fails as we enter a resonance.
The oscillatory terms in the equations of motion combine in phase on a resonance,
“kicking” the system until it evolves away from the resonance. The adiabatic evo-
lutiondoeswellatdescribingthesystem’sevolutionbeforeandaftertheresonance,
butdoesapoorjobatmodelinginthe systemverynearthe resonance. Takingres-
onances into account, we find that the system’s phase evolution must be modified:
t2
Φ(t ,t )= ω(t)dt
1 2 Z
t1
= Φdiss−1+ΦRES+Φcons−1+Φdiss−2+Φcons−2+... . (18)
This form is identical to that given earlier, but there is now a new term: Φ =
RES
O(ǫ1/2T/T ) = O([M/m]1/2). This term dominates over all but the leading dissi-
i o
pative contributions to the system’s phase evolution. Detailed analysis10,11 indeed
shows that Φ contributes dozens to hundreds of radians to the system’s phase
RES
evolution, substantially more than all terms except the leading one.
One might imagine that, since resonant orbits are a set of measure zero in the
completesetofKerrblackholeorbits,thesecasesarecuriositiesthatareunlikelyto
playmuchroleinastrophysics. Thatisnotthecase. Considerasetofastrophysical
inspirals withparameterssuchthatthey arelikely to be importantsourcesforlow-
frequency GW detector. We have shown11 that every inspiral will pass through
at least one dynamically significant low-order resonance as it spirals through the
detector’s sensitive band. Many of these inspirals will pass through two significant
resonances; some will pass through three.
5. Summary and outlook
Although useful for producing a somewhat accurate picture of extreme mass ratio
binaries, the adiabatic approach to inspiral is ultimately inadequate for modeling
thesesources,evenforthelessstringenttaskofdevelopingdetectiontemplates. We
must go beyond this picture and develop post-adiabatic EMRI models in order to
more completely model these sources.
Of particular importance is understanding the magnitude of the “kick” that is
imparted to an EMRI’s evolution by each resonance passage. Properly doing this
requires that we self consistently integrate the equations of motion, including the
oscillatory part of the self interaction. Past work9–11 has given us some idea how
much kick we can expect and the number of cycles over which the kick operates,
but it remains important to develop a fully self consistent inspiral and waveform
model to assess the reliability of these estimates.
Even with good modeling, it is likely that the impact of resonances will sub-
stantially complicate our ability to measure EMRI GWs. The detailed evolution
of an EMRI on resonance depends on the value of two orbital phases as we enter
resonance. Fornon-resonantorbits,thesephasesareignorable. Resonancesthusin-
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crease the dimensionality of the EMRI waveformmanifold, and potentially greatly
expand the number of parameters that will be needed for measurement templates.
For detection purposes at least, it may be adequate to break the data into
segments. A simple, adiabatic model suffices to model the system’s phase until
we are within a few dozen or hundred radians of the resonance; a different simple,
adiabatic model suffices to model the phase once we are a few dozen or hundred
radians past resonance. Each EMRI is thus broken into pre- and post-resonance
segments. Similartechniquesareusedinradioastronomytomodelglitchingpulsars,
when a pulsar’s spin frequency suddenly changes following a change in a neutron
star’smomentofinertiaduetoarearrangementofitsinternalfluiddistribution(see,
e.g., Fig. 1 of Ref. 12 for an example and discussion). This segmented approach is
likely to work well on EMRI events whose fully coherent signal-to-noise ratios are
several tens or larger.
Much work remains to be done on this challenging problem.
Acknowledgments
Research in our group is supported by National Science Grant PHY-1403261. The
work discussed here has been done in collaboration with E´anna Flanagan and
Uchupol Ruangsri, and has benefitted greatly from discussions with Tanja Hin-
derer, Eric Poisson,and Niels Warburton.
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